[✓] Use transformation matrices that act on vector?

Hi ,
I'm new with Mathematica and would like to know how to write transformation matrices (operators) that act on vectors (for tensor analysis). Or a Jacobinan-type matrix For example, a differential operator that acts on a column vector of functions:
row 1: d[ ]/dx1 d[ ]/dx2
row 2: d[ ]/dx3 d[ ]/dx4

How can I do that in Mathematica? The regular linear algebra functions don't work.
AUSP6